1034:
1099:
1781:
2873:
1788:
1442:
1804:
1449:
296:
478:
232:
359:
1811:
1774:
1620:
1613:
770:
414:
749:
1738:
1673:
1567:
1920:
1560:
2907:
2866:
1795:
1724:
1717:
1428:
2900:
2859:
1927:
1731:
1708:
1657:
1435:
1395:
1381:
1650:
1643:
1507:
1402:
1388:
1345:
289:
1500:
1336:
471:
352:
225:
1701:
1537:
1493:
1479:
1472:
791:
1694:
1687:
1606:
1599:
1551:
1544:
1523:
1486:
1463:
1456:
1367:
1836:
1767:
1636:
1629:
1530:
1516:
1374:
1664:
407:
985:
1829:
1680:
936:
887:
1248:
1220:
1760:
1592:
1585:
1322:
1315:
1308:
1301:
1241:
1234:
1227:
1213:
2914:
1329:
1292:
1285:
1278:
94:
1271:
1264:
1257:
2852:
2780:
There are eighteen two-parameter families of regular compound tessellations of the
Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not been enumerated.
837:
of both solids. The hull is the dual of this rectification, and its rhombic faces have the intersecting edges of the two solids as diagonals (and have their four alternate vertices). For the convex solids, this is the
1195:
The 75 uniform compounds are listed in the Table below. Most are shown singularly colored by each polyhedron element. Some chiral pairs of face groups are colored by symmetry of the faces within each polyhedron.
1025:
1090:
565:
Each of the regular tetrahedral compounds is self-dual or dual to its chiral twin; the regular compound of five cubes and the regular compound of five octahedra are dual to each other.
1843:
The compound of four cubes (left) is neither a regular compound, nor a dual compound, nor a uniform compound. Its dual, the compound of four octahedra (right), is a uniform compound.
562:
versions, which together make up the regular compound of ten tetrahedra. The regular compound of ten tetrahedra can also be seen as a compound of five stellae octangulae.
3276:
2319:
The superscript (var) in the tables above indicates that the labeled compounds are distinct from the other compounds with the same number of constituents.
1164:
The small stellated dodecahedral (or great dodecahedral) dual compound has the great dodecahedron completely interior to the small stellated dodecahedron.
76:
Another convex polyhedron is formed by the small central space common to all members of the compound. This polyhedron can be used as the core for a set of
2691:
976:
2709:
2655:
830:, such that the edge of one polyhedron intersects the dual edge of the dual polyhedron. There are five dual compounds of the regular polyhedra.
3391:
2673:
1355:
3174:
1191:
3363:
3348:
3019:
2889:
2843:
2609:
2591:
1893:
860:
195:
133:
170:; the compound of two tetrahedra is the only regular compound with that property. There are five regular compounds of polyhedra:
2926:, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs.
2641:
1874:
1862:
927:
3184:
3005:
2714:
2526:
2509:
1039:
1033:
3410:
3043:
2942:
2922:
A known family of regular
Euclidean compound honeycombs in any number of dimensions is an infinite family of compounds of
1104:
115:
104:
1901:
1878:
1084:
1015:
1850:
1173:
779:
555:
280:
2627:
2700:
1780:
874:
462:
343:
216:
2872:
1098:
3339:
2682:
2664:
2646:
2377:
2366:
1954:
834:
2745:
on the polyhedra (so that each polyhedron can be sent to any of the others, as in uniform compounds), then if
2576:
1787:
1441:
1190:
and every vertex is transitive with every other vertex.) This list includes the five regular compounds above.
2561:
1803:
1448:
1117:
990:
800:
559:
424:
336:
331:
3167:
295:
1855:
477:
398:
358:
231:
111:
3189:
754:
1810:
1773:
1619:
1612:
3225:
2923:
2696:
2660:
1897:
1416:
1147:
1110:
1058:
966:
941:
878:
3299:
769:
413:
3201:
2981:
2750:
1737:
1672:
1566:
1412:
1186:
sets of compounds, #20-#25) made from uniform polyhedra with rotational symmetry. (Every vertex is
1123:
1045:
970:
758:
3161:
3139:
1919:
1559:
748:
3249:
3124:
1886:
1870:
1146:
The tetrahedron is self-dual, so the dual compound of a tetrahedron with its dual is the regular
1052:
1019:
147:
2906:
2865:
1794:
1723:
1716:
1427:
576:
2899:
2858:
1926:
1730:
1707:
1656:
1434:
1394:
1380:
3387:
3359:
3344:
3270:
3025:
3015:
2938:
2837:
2742:
2485:
2199:
1882:
1649:
1642:
1506:
1401:
1387:
1344:
1187:
1158:
1080:
995:
810:
796:
547:, which shares the same face-planes as the compound. Thus the compound of two tetrahedra is a
483:
288:
151:
62:
58:
3379:
3285:
3233:
2934:
2831:
2801:
1499:
1335:
1183:
532:
470:
351:
224:
3245:
3179:
1892:
The section for enantiomorph pairs in
Skilling's list does not contain the compound of two
1861:
Two polyhedra that are compounds but have their elements rigidly locked into place are the
1700:
1536:
1492:
1478:
1471:
3241:
2945:. The Euclidean compounds of two hypercubic honeycombs are both regular and dual-regular.
2492:
2206:
1693:
1686:
1605:
1598:
1550:
1543:
1522:
1485:
1462:
1455:
1366:
540:
536:
237:
167:
159:
155:
3229:
1835:
1766:
1635:
1628:
1529:
1515:
1373:
826:
compound is composed of a polyhedron and its dual, arranged reciprocally about a common
790:
3315:
Schriften der
Gesellschaft zur Berörderung der Gasammten Naturwissenschaften zu Marburg
3156:
2718:
2678:
2632:
2520:
2503:
2355:
2344:
1958:
1663:
406:
163:
3404:
3253:
2825:
1949:
In 4-dimensions, there are a large number of regular compounds of regular polytopes.
1154:
946:
814:
579:. The material inside the square brackets, , denotes the components of the compound:
984:
575:
Coxeter's notation for regular compounds is given in the table above, incorporating
3327:
2734:
1828:
1748:
1679:
935:
886:
301:
47:
17:
1247:
1219:
735:
times. This notation can be generalised to compounds in any number of dimensions.
3383:
3009:
3310:
2754:
1866:
1759:
1591:
1584:
1321:
1314:
1307:
1300:
1240:
1233:
1226:
1212:
868:
839:
528:
308:
187:
66:
3195:
Compound of Small
Stellated Dodecahedron and Great Dodecahedron {5/2,5}+{5,5/2}
2913:
1328:
1291:
1284:
1277:
1153:
The octahedral and icosahedral dual compounds are the first stellations of the
3237:
921:
897:
548:
544:
244:
77:
3194:
2596:
2223:
2163:
2146:
2129:
2112:
2095:
2078:
1935:
827:
775:
70:
39:
1576:
46-67: Tetrahedral symmetry embedded in octahedral or icosahedral symmetry,
3029:
1270:
1263:
1256:
2618:
2614:
2291:
2285:
2274:
2268:
200:
146:
A regular polyhedral compound can be defined as a compound which, like a
51:
43:
31:
2753:
of a single chosen polyhedron, the polyhedra can be identified with the
3378:, Bolyai Society Mathematical Studies, vol. 27, pp. 307–320,
2600:
2581:
2308:
2302:
2257:
2251:
2240:
2234:
2217:
2180:
2174:
2157:
2140:
2123:
2106:
2089:
2072:
2041:
2036:
1950:
1941:
3190:
http://users.skynet.be/polyhedra.fleurent/Compounds_2/Compounds_2.htm
2566:
2025:
2020:
2009:
2004:
1993:
1988:
595:
the square brackets denotes the vertex arrangement of the compound:
551:
of the octahedron, and in fact, the only finite stellation thereof.
2851:
1900:
faces would coincide. Removing the coincident faces results in the
635:
the square brackets denotes the facet arrangement of the compound:
1823:
3216:
Skilling, John (1976), "Uniform
Compounds of Uniform Polyhedra",
3374:
McMullen, Peter (2018), "New
Regular Compounds of 4-Polytopes",
3292:, Cambridge, England: Cambridge University Press, pp. 51–53
917:
892:
3313:(1876), "Zugleich Gleicheckigen und Gleichflächigen Polyeder",
3218:
Mathematical
Proceedings of the Cambridge Philosophical Society
2741:
is the symmetry group of a polyhedral compound, and the group
162:. Unlike the case of polyhedra, this is not equivalent to the
87:
2808:
A few examples of
Euclidean and hyperbolic regular compounds
568:
Hence, regular polyhedral compounds can also be regarded as
3044:"Great Dodecahedron-Small Stellated Dodecahedron Compound"
2933:
tiling compounds. A simple example is the E compound of a
3358:. California: University of California Press Berkeley.
1182:
which enumerated 75 compounds (including 6 as infinite
1911:
2191:
Uniform compounds and duals with convex 4-polytopes:
114:. Please help to ensure that disputed statements are
3175:
Skilling's 75 Uniform
Compounds of Uniform Polyhedra
2692:
1 great grand 120-cell, 1 great icosahedral 120-cell
543:, and the intersection of the two define a regular
3014:(Third ed.). Dover Publications. p. 48.
2784:The Euclidean and hyperbolic compound families 2 {
2710:1 great grand stellated 120-cell, 1 grand 600-cell
2656:1 icosahedral 120-cell, 1 small stellated 120-cell
27:3D shape made of polyhedra sharing a common center
3180:Skilling's Uniform Compounds of Uniform Polyhedra
3113:, New Trends in Intuitive Geometry, 27: 307–320
539:. The vertices of the two tetrahedra define a
3125:"Uniform compound stellated icositetrachoron"
46:. They are the three-dimensional analogs of
8:
2674:1 grand 120-cell, 1 great stellated 120-cell
2800:an integer) are analogous to the spherical
3275:: CS1 maint: location missing publisher (
3105:
3103:
3101:
3099:
1200:1-19: Miscellaneous (4,5,6,9,17 are the 5
527:Best known is the regular compound of two
134:Learn how and when to remove this message
61:of a compound can be connected to form a
2806:
2546:
2479:
2392:
2331:
2193:
2056:
1972:
977:Compound of dodecahedron and icosahedron
844:
172:
110:Relevant discussion may be found on the
38:is a figure that is composed of several
3164:– from Virtual Reality Polyhedra
2954:
2323:Compounds with regular star 4-polytopes
3343:, (3rd edition, 1973), Dover edition,
3268:
3168:Uniform Compounds of Uniform Polyhedra
3089:
3087:
3085:
3083:
3081:
3079:
3077:
3075:
3073:
1180:Uniform Compounds of Uniform Polyhedra
3071:
3069:
3067:
3065:
3063:
3061:
3059:
3057:
3055:
3053:
7:
3111:New Regular Compounds of 4-Polytopes
3093:Regular polytopes, Table VII, p. 305
2976:
2974:
2972:
2970:
2968:
2966:
2964:
2962:
2960:
2958:
1963:New Regular Compounds of 4-Polytopes
3140:"Uniform compound demidistesseract"
671:times. These may be combined: thus
2941:, which shares its edges with the
1889:is generalised, they are uniform.
1411:26-45: Prism symmetry embedded in
1354:20-25: Prism symmetry embedded in
25:
3370:p. 87 Five regular compounds
1953:lists a few of these in his book
3376:New Trends in Intuitive Geometry
2912:
2905:
2898:
2871:
2864:
2857:
2850:
2772:sends the chosen polyhedron to.
2768:corresponds to which polyhedron
2475:Uniform compound stars and duals
1925:
1918:
1894:great snub dodecicosidodecahedra
1834:
1827:
1809:
1802:
1793:
1786:
1779:
1772:
1765:
1758:
1736:
1729:
1722:
1715:
1706:
1699:
1692:
1685:
1678:
1671:
1662:
1655:
1648:
1641:
1634:
1627:
1618:
1611:
1604:
1597:
1590:
1583:
1565:
1558:
1549:
1542:
1535:
1528:
1521:
1514:
1505:
1498:
1491:
1484:
1477:
1470:
1461:
1454:
1447:
1440:
1433:
1426:
1400:
1393:
1386:
1379:
1372:
1365:
1343:
1334:
1327:
1320:
1313:
1306:
1299:
1290:
1283:
1276:
1269:
1262:
1255:
1246:
1239:
1232:
1225:
1218:
1211:
1178:In 1976 John Skilling published
1097:
1032:
983:
934:
885:
789:
768:
747:
476:
469:
412:
405:
357:
350:
294:
287:
230:
223:
92:
3006:Coxeter, Harold Scott MacDonald
1875:great complex icosidodecahedron
1863:small complex icosidodecahedron
928:Compound of cube and octahedron
3157:MathWorld: Polyhedron Compound
2715:great grand stellated 120-cell
1040:Medial rhombic triacontahedron
1:
2943:deltoidal trihexagonal tiling
2389:Dual pairs of compound stars:
1105:Great rhombic triacontahedron
707:}'s sharing the vertices of {
619:}'s sharing the vertices of {
3384:10.1007/978-3-662-57413-3_12
3356:Polyhedra: A visual approach
2848:
1902:compound of twenty octahedra
1879:small stellated dodecahedron
1085:great stellated dodecahedron
1016:Small stellated dodecahedron
460:
396:
341:
278:
214:
3298:Harman, Michael G. (1974),
3261:Cromwell, Peter R. (1997),
1851:Compound of three octahedra
1174:Uniform polyhedron compound
780:pentagonal icositetrahedron
556:compound of five tetrahedra
166:acting transitively on its
3427:
2701:great icosahedral 120-cell
2642:2 grand stellated 120-cell
1885:). If the definition of a
1171:
875:Compound of two tetrahedra
659:}'s sharing the faces of {
3238:10.1017/S0305004100052440
2815:
2328:Self-dual star compounds:
1842:
3304:, unpublished manuscript
3109:McMullen, Peter (2018),
2683:great stellated 120-cell
2665:small stellated 120-cell
2647:grand stellated 120-cell
535:, a name given to it by
1961:added six in his paper
1913:Orthogonal projections
1118:Great icosidodecahedron
991:Rhombic triacontahedron
801:rhombic triacontahedron
425:Rhombic triacontahedron
2610:1 120-cell, 1 600-cell
2592:1 tesseract, 1 16-cell
1856:Compound of four cubes
1091:Compound of gI and gsD
570:dual-regular compounds
211:Dual-regular compound
3354:Anthony Pugh (1976).
3332:De Divina Proportione
2924:hypercubic honeycombs
1026:Compound of sD and gD
755:Truncated tetrahedron
3411:Polyhedral compounds
3301:Polyhedral Compounds
3202:"Compound polytopes"
3185:Polyhedral Compounds
2982:"Compound Polyhedra"
2776:Compounds of tilings
2697:great grand 120-cell
2661:icosahedral 120-cell
2538:Compounds with duals
1908:4-polytope compounds
1417:icosahedral symmetry
1148:stellated octahedron
942:Rhombic dodecahedron
879:stellated octahedron
695:} is a compound of
631:times. The material
103:factual accuracy is
73:of its convex hull.
3230:1976MPCPS..79..447S
3200:Klitzing, Richard.
3138:Klitzing, Richard.
3123:Klitzing, Richard.
2809:
1914:
759:triakis tetrahedron
647:} is a compound of
607:} is a compound of
531:, often called the
48:polygonal compounds
36:polyhedral compound
18:Polyhedron compound
3162:Compound polyhedra
2986:www.georgehart.com
2807:
1912:
1887:uniform polyhedron
1871:great dodecahedron
1053:Dodecadodecahedron
1020:great dodecahedron
809:Dual compounds of
591:}'s. The material
148:regular polyhedron
69:. A compound is a
3393:978-3-662-57412-6
3340:Regular Polytopes
3286:Wenninger, Magnus
3011:Regular Polytopes
2939:triangular tiling
2920:
2919:
2743:acts transitively
2726:
2725:
2535:
2534:
2486:Vertex-transitive
2472:
2471:
2386:
2385:
2317:
2316:
2200:Vertex-transitive
2189:
2188:
2050:
2049:
1955:Regular Polytopes
1947:
1946:
1883:great icosahedron
1847:
1846:
1817:
1816:
1744:
1743:
1573:
1572:
1408:
1407:
1351:
1350:
1202:regular compounds
1188:vertex-transitive
1168:Uniform compounds
1159:icosidodecahedron
1144:
1143:
1127:
1114:
1094:
1081:Great icosahedron
1062:
1049:
1029:
996:Icosidodecahedron
980:
931:
882:
797:Icosidodecahedron
525:
524:
484:Icosidodecahedron
178:(Coxeter symbol)
152:vertex-transitive
144:
143:
136:
84:Regular compounds
63:convex polyhedron
42:sharing a common
16:(Redirected from
3418:
3396:
3369:
3334:
3322:
3305:
3293:
3280:
3274:
3266:
3256:
3205:
3144:
3143:
3135:
3129:
3128:
3120:
3114:
3107:
3094:
3091:
3048:
3047:
3040:
3034:
3033:
3002:
2996:
2995:
2993:
2992:
2978:
2935:hexagonal tiling
2916:
2909:
2902:
2875:
2868:
2861:
2854:
2810:
2802:stella octangula
2628:2 great 120-cell
2547:
2480:
2393:
2332:
2194:
2057:
1973:
1929:
1922:
1915:
1838:
1831:
1824:
1813:
1806:
1797:
1790:
1783:
1776:
1769:
1762:
1755:
1754:
1740:
1733:
1726:
1719:
1710:
1703:
1696:
1689:
1682:
1675:
1666:
1659:
1652:
1645:
1638:
1631:
1622:
1615:
1608:
1601:
1594:
1587:
1580:
1579:
1569:
1562:
1553:
1546:
1539:
1532:
1525:
1518:
1509:
1502:
1495:
1488:
1481:
1474:
1465:
1458:
1451:
1444:
1437:
1430:
1423:
1422:
1404:
1397:
1390:
1383:
1376:
1369:
1362:
1361:
1347:
1338:
1331:
1324:
1317:
1310:
1303:
1294:
1287:
1280:
1273:
1266:
1259:
1250:
1243:
1236:
1229:
1222:
1215:
1208:
1207:
1161:, respectively.
1121:
1108:
1101:
1088:
1056:
1043:
1036:
1023:
987:
974:
938:
925:
889:
872:
845:
833:The core is the
793:
772:
751:
577:Schläfli symbols
533:stella octangula
480:
473:
416:
409:
361:
354:
298:
291:
234:
227:
176:Regular compound
173:
139:
132:
128:
125:
119:
116:reliably sourced
96:
95:
88:
21:
3426:
3425:
3421:
3420:
3419:
3417:
3416:
3415:
3401:
3400:
3394:
3373:
3366:
3353:
3326:
3309:
3297:
3284:
3267:
3260:
3215:
3212:
3199:
3153:
3148:
3147:
3137:
3136:
3132:
3122:
3121:
3117:
3108:
3097:
3092:
3051:
3042:
3041:
3037:
3022:
3004:
3003:
2999:
2990:
2988:
2980:
2979:
2956:
2951:
2929:There are also
2778:
2731:
2543:Dual positions:
2540:
2493:Cell-transitive
2491:
2484:
2325:
2207:Cell-transitive
2205:
2198:
1910:
1822:
1820:Other compounds
1176:
1170:
1140:
1132:
1131:
1120:
1107:
1087:
1075:
1067:
1066:
1055:
1042:
1022:
1010:
1002:
1001:
973:
961:
953:
952:
924:
912:
904:
903:
871:
820:
819:
818:
817:
806:
805:
804:
794:
785:
784:
783:
773:
764:
763:
762:
752:
741:
518:
510:
509:
505:
497:
496:
465:
457:Five octahedra
454:
446:
445:
441:
433:
432:
401:
393:Ten tetrahedra
387:
386:
382:
374:
373:
346:
335:
325:
324:
317:
316:
283:
281:Five tetrahedra
275:Two tetrahedra
272:
264:
263:
259:
251:
250:
219:
207:
205:
203:
177:
160:face-transitive
156:edge-transitive
140:
129:
123:
120:
109:
101:This section's
97:
93:
86:
28:
23:
22:
15:
12:
11:
5:
3424:
3422:
3414:
3413:
3403:
3402:
3399:
3398:
3392:
3371:
3364:
3351:
3336:
3324:
3307:
3295:
3282:
3258:
3224:(3): 447–457,
3211:
3208:
3207:
3206:
3197:
3192:
3187:
3182:
3177:
3172:
3171:
3170:
3159:
3152:
3151:External links
3149:
3146:
3145:
3130:
3115:
3095:
3049:
3035:
3020:
2997:
2953:
2952:
2950:
2947:
2918:
2917:
2910:
2903:
2896:
2893:
2892:
2886:
2883:
2880:
2877:
2876:
2869:
2862:
2855:
2847:
2846:
2840:
2834:
2828:
2821:
2820:
2817:
2814:
2777:
2774:
2730:
2727:
2724:
2723:
2721:
2719:grand 600-cell
2712:
2706:
2705:
2703:
2694:
2688:
2687:
2685:
2679:grand 120-cell
2676:
2670:
2669:
2667:
2658:
2652:
2651:
2649:
2644:
2638:
2637:
2635:
2633:great 120-cell
2630:
2624:
2623:
2621:
2612:
2606:
2605:
2603:
2594:
2588:
2587:
2586:], order 2304
2584:
2579:
2573:
2572:
2569:
2564:
2558:
2557:
2554:
2551:
2539:
2536:
2533:
2532:
2531:, order 14400
2529:
2523:
2516:
2515:
2512:
2506:
2499:
2498:
2495:
2488:
2470:
2469:
2468:, order 14400
2466:
2463:
2459:
2458:
2455:
2452:
2448:
2447:
2446:, order 14400
2444:
2441:
2437:
2436:
2433:
2430:
2426:
2425:
2424:, order 14400
2422:
2419:
2415:
2414:
2411:
2408:
2404:
2403:
2400:
2397:
2384:
2383:
2382:, order 14400
2380:
2373:
2372:
2369:
2362:
2361:
2360:, order 14400
2358:
2351:
2350:
2347:
2340:
2339:
2336:
2324:
2321:
2315:
2314:
2311:
2305:
2298:
2297:
2296:, order 14400
2294:
2288:
2281:
2280:
2277:
2271:
2264:
2263:
2262:, order 14400
2260:
2254:
2247:
2246:
2243:
2237:
2230:
2229:
2226:
2220:
2213:
2212:
2209:
2202:
2187:
2186:
2185:, order 14400
2183:
2177:
2170:
2169:
2168:, order 14400
2166:
2160:
2153:
2152:
2149:
2143:
2136:
2135:
2132:
2126:
2119:
2118:
2117:, order 14400
2115:
2109:
2102:
2101:
2100:, order 14400
2098:
2092:
2085:
2084:
2081:
2075:
2068:
2067:
2064:
2061:
2048:
2047:
2046:, order 14400
2044:
2039:
2032:
2031:
2030:, order 14400
2028:
2023:
2016:
2015:
2012:
2007:
2000:
1999:
1998:, order 14400
1996:
1991:
1984:
1983:
1980:
1977:
1945:
1944:
1938:
1931:
1930:
1923:
1909:
1906:
1859:
1858:
1853:
1845:
1844:
1840:
1839:
1832:
1821:
1818:
1815:
1814:
1807:
1799:
1798:
1791:
1784:
1777:
1770:
1763:
1753:
1752:
1742:
1741:
1734:
1727:
1720:
1712:
1711:
1704:
1697:
1690:
1683:
1676:
1668:
1667:
1660:
1653:
1646:
1639:
1632:
1624:
1623:
1616:
1609:
1602:
1595:
1588:
1578:
1577:
1571:
1570:
1563:
1555:
1554:
1547:
1540:
1533:
1526:
1519:
1511:
1510:
1503:
1496:
1489:
1482:
1475:
1467:
1466:
1459:
1452:
1445:
1438:
1431:
1421:
1420:
1406:
1405:
1398:
1391:
1384:
1377:
1370:
1360:
1359:
1356:prism symmetry
1349:
1348:
1340:
1339:
1332:
1325:
1318:
1311:
1304:
1296:
1295:
1288:
1281:
1274:
1267:
1260:
1252:
1251:
1244:
1237:
1230:
1223:
1216:
1206:
1205:
1172:Main article:
1169:
1166:
1142:
1141:
1136:
1128:
1115:
1102:
1095:
1077:
1076:
1071:
1063:
1050:
1037:
1030:
1012:
1011:
1006:
998:
993:
988:
981:
963:
962:
957:
949:
944:
939:
932:
914:
913:
908:
900:
895:
890:
883:
864:
863:
861:Symmetry group
858:
855:
852:
849:
848:Dual compound
815:Catalan solids
808:
807:
795:
788:
787:
786:
774:
767:
766:
765:
753:
746:
745:
744:
743:
742:
740:
739:Dual compounds
737:
723:the faces of {
560:enantiomorphic
523:
522:
519:
514:
506:
501:
493:
488:
481:
474:
467:
463:Five octahedra
459:
458:
455:
450:
442:
437:
429:
422:
417:
410:
403:
395:
394:
391:
383:
378:
370:
367:
362:
355:
348:
344:Ten tetrahedra
340:
339:
337:(Enantiomorph)
329:
321:
313:
306:
299:
292:
285:
277:
276:
273:
268:
260:
255:
247:
242:
235:
228:
221:
217:Two tetrahedra
213:
212:
209:
198:
196:Symmetry group
193:
190:
185:
182:
179:
164:symmetry group
142:
141:
100:
98:
91:
85:
82:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3423:
3412:
3409:
3408:
3406:
3395:
3389:
3385:
3381:
3377:
3372:
3367:
3365:0-520-03056-7
3361:
3357:
3352:
3350:
3349:0-486-61480-8
3346:
3342:
3341:
3337:
3333:
3329:
3328:Pacioli, Luca
3325:
3320:
3316:
3312:
3308:
3303:
3302:
3296:
3291:
3287:
3283:
3278:
3272:
3264:
3259:
3255:
3251:
3247:
3243:
3239:
3235:
3231:
3227:
3223:
3219:
3214:
3213:
3209:
3203:
3198:
3196:
3193:
3191:
3188:
3186:
3183:
3181:
3178:
3176:
3173:
3169:
3166:
3165:
3163:
3160:
3158:
3155:
3154:
3150:
3141:
3134:
3131:
3126:
3119:
3116:
3112:
3106:
3104:
3102:
3100:
3096:
3090:
3088:
3086:
3084:
3082:
3080:
3078:
3076:
3074:
3072:
3070:
3068:
3066:
3064:
3062:
3060:
3058:
3056:
3054:
3050:
3045:
3039:
3036:
3031:
3027:
3023:
3021:0-486-61480-8
3017:
3013:
3012:
3007:
3001:
2998:
2987:
2983:
2977:
2975:
2973:
2971:
2969:
2967:
2965:
2963:
2961:
2959:
2955:
2948:
2946:
2944:
2940:
2937:and its dual
2936:
2932:
2927:
2925:
2915:
2911:
2908:
2904:
2901:
2897:
2895:
2894:
2891:
2887:
2884:
2881:
2879:
2878:
2874:
2870:
2867:
2863:
2860:
2856:
2853:
2849:
2845:
2841:
2839:
2835:
2833:
2829:
2827:
2823:
2822:
2818:
2812:
2811:
2805:
2803:
2799:
2795:
2791:
2787:
2782:
2775:
2773:
2771:
2767:
2763:
2759:
2756:
2752:
2748:
2744:
2740:
2736:
2728:
2722:
2720:
2716:
2713:
2711:
2708:
2707:
2704:
2702:
2698:
2695:
2693:
2690:
2689:
2686:
2684:
2680:
2677:
2675:
2672:
2671:
2668:
2666:
2662:
2659:
2657:
2654:
2653:
2650:
2648:
2645:
2643:
2640:
2639:
2636:
2634:
2631:
2629:
2626:
2625:
2622:
2620:
2616:
2613:
2611:
2608:
2607:
2604:
2602:
2598:
2595:
2593:
2590:
2589:
2585:
2583:
2580:
2578:
2575:
2574:
2571:], order 240
2570:
2568:
2565:
2563:
2560:
2559:
2555:
2552:
2549:
2548:
2545:
2544:
2537:
2530:
2528:
2524:
2522:
2518:
2517:
2514:, order 7200
2513:
2511:
2507:
2505:
2501:
2500:
2496:
2494:
2489:
2487:
2482:
2481:
2478:
2476:
2467:
2464:
2461:
2460:
2457:, order 7200
2456:
2453:
2450:
2449:
2445:
2442:
2439:
2438:
2435:, order 7200
2434:
2431:
2428:
2427:
2423:
2420:
2417:
2416:
2413:, order 7200
2412:
2409:
2406:
2405:
2401:
2398:
2395:
2394:
2391:
2390:
2381:
2379:
2375:
2374:
2371:, order 7200
2370:
2368:
2364:
2363:
2359:
2357:
2353:
2352:
2349:, order 7200
2348:
2346:
2342:
2341:
2337:
2334:
2333:
2330:
2329:
2322:
2320:
2312:
2310:
2306:
2304:
2300:
2299:
2295:
2293:
2289:
2287:
2283:
2282:
2279:, order 7200
2278:
2276:
2272:
2270:
2266:
2265:
2261:
2259:
2255:
2253:
2249:
2248:
2245:, order 7200
2244:
2242:
2238:
2236:
2232:
2231:
2227:
2225:
2221:
2219:
2215:
2214:
2210:
2208:
2203:
2201:
2196:
2195:
2192:
2184:
2182:
2178:
2176:
2172:
2171:
2167:
2165:
2161:
2159:
2155:
2154:
2151:, order 7200
2150:
2148:
2144:
2142:
2138:
2137:
2133:
2131:
2127:
2125:
2121:
2120:
2116:
2114:
2110:
2108:
2104:
2103:
2099:
2097:
2093:
2091:
2087:
2086:
2083:, order 1152
2082:
2080:
2076:
2074:
2070:
2069:
2065:
2062:
2059:
2058:
2055:
2054:
2045:
2043:
2040:
2038:
2034:
2033:
2029:
2027:
2024:
2022:
2018:
2017:
2013:
2011:
2008:
2006:
2002:
2001:
1997:
1995:
1992:
1990:
1986:
1985:
1981:
1978:
1975:
1974:
1971:
1970:
1966:
1964:
1960:
1956:
1952:
1943:
1939:
1937:
1933:
1932:
1928:
1924:
1921:
1917:
1916:
1907:
1905:
1903:
1899:
1895:
1890:
1888:
1884:
1880:
1877:(compound of
1876:
1872:
1868:
1865:(compound of
1864:
1857:
1854:
1852:
1849:
1848:
1841:
1837:
1833:
1830:
1826:
1825:
1819:
1812:
1808:
1805:
1801:
1800:
1796:
1792:
1789:
1785:
1782:
1778:
1775:
1771:
1768:
1764:
1761:
1757:
1756:
1750:
1746:
1745:
1739:
1735:
1732:
1728:
1725:
1721:
1718:
1714:
1713:
1709:
1705:
1702:
1698:
1695:
1691:
1688:
1684:
1681:
1677:
1674:
1670:
1669:
1665:
1661:
1658:
1654:
1651:
1647:
1644:
1640:
1637:
1633:
1630:
1626:
1625:
1621:
1617:
1614:
1610:
1607:
1603:
1600:
1596:
1593:
1589:
1586:
1582:
1581:
1575:
1574:
1568:
1564:
1561:
1557:
1556:
1552:
1548:
1545:
1541:
1538:
1534:
1531:
1527:
1524:
1520:
1517:
1513:
1512:
1508:
1504:
1501:
1497:
1494:
1490:
1487:
1483:
1480:
1476:
1473:
1469:
1468:
1464:
1460:
1457:
1453:
1450:
1446:
1443:
1439:
1436:
1432:
1429:
1425:
1424:
1418:
1414:
1410:
1409:
1403:
1399:
1396:
1392:
1389:
1385:
1382:
1378:
1375:
1371:
1368:
1364:
1363:
1357:
1353:
1352:
1346:
1342:
1341:
1337:
1333:
1330:
1326:
1323:
1319:
1316:
1312:
1309:
1305:
1302:
1298:
1297:
1293:
1289:
1286:
1282:
1279:
1275:
1272:
1268:
1265:
1261:
1258:
1254:
1253:
1249:
1245:
1242:
1238:
1235:
1231:
1228:
1224:
1221:
1217:
1214:
1210:
1209:
1203:
1199:
1198:
1197:
1193:
1192:
1189:
1185:
1181:
1175:
1167:
1165:
1162:
1160:
1156:
1155:cuboctahedron
1151:
1149:
1139:
1135:
1129:
1125:
1119:
1116:
1112:
1106:
1103:
1100:
1096:
1092:
1086:
1082:
1079:
1078:
1074:
1070:
1064:
1060:
1054:
1051:
1047:
1041:
1038:
1035:
1031:
1027:
1021:
1017:
1014:
1013:
1009:
1005:
999:
997:
994:
992:
989:
986:
982:
978:
972:
968:
965:
964:
960:
956:
950:
948:
947:Cuboctahedron
945:
943:
940:
937:
933:
929:
923:
919:
916:
915:
911:
907:
901:
899:
896:
894:
891:
888:
884:
880:
876:
870:
866:
865:
862:
859:
856:
853:
850:
847:
846:
843:
841:
836:
835:rectification
831:
829:
825:
816:
812:
802:
798:
792:
781:
777:
771:
760:
756:
750:
738:
736:
734:
730:
726:
722:
718:
714:
710:
706:
702:
698:
694:
690:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
646:
642:
638:
634:
630:
626:
622:
618:
614:
610:
606:
602:
598:
594:
590:
586:
582:
578:
573:
571:
566:
563:
561:
558:comes in two
557:
552:
550:
546:
542:
538:
534:
530:
520:
517:
513:
507:
504:
500:
494:
492:
489:
487:
485:
482:
479:
475:
472:
468:
464:
461:
456:
453:
449:
443:
440:
436:
430:
428:
426:
423:
421:
419:Dodecahedron
418:
415:
411:
408:
404:
400:
397:
392:
390:
384:
381:
377:
371:
368:
366:
364:Dodecahedron
363:
360:
356:
353:
349:
347:2{5,3}2{3,5}
345:
342:
338:
333:
330:
328:
322:
320:
314:
312:
310:
307:
305:
303:
300:
297:
293:
290:
286:
282:
279:
274:
271:
267:
261:
258:
254:
248:
246:
243:
241:
239:
236:
233:
229:
226:
222:
218:
215:
210:
202:
199:
197:
194:
191:
189:
186:
183:
180:
175:
174:
171:
169:
165:
161:
157:
153:
149:
138:
135:
127:
124:November 2023
117:
113:
107:
106:
99:
90:
89:
83:
81:
79:
74:
72:
68:
64:
60:
55:
53:
49:
45:
41:
37:
33:
19:
3375:
3355:
3338:
3331:
3318:
3314:
3311:Hess, Edmund
3300:
3289:
3262:
3221:
3217:
3133:
3118:
3110:
3038:
3010:
3000:
2989:. Retrieved
2985:
2931:dual-regular
2930:
2928:
2921:
2797:
2793:
2789:
2785:
2783:
2779:
2769:
2765:
2764:– the coset
2761:
2757:
2746:
2738:
2735:group theory
2733:In terms of
2732:
2729:Group theory
2553:Constituent
2542:
2541:
2474:
2473:
2465:10 {5,3,5/2}
2462:10 {5/2,3,5}
2443:10 {3,5/2,5}
2440:10 {5,5/2,3}
2421:10 {5/2,5,3}
2418:10 {3,5,5/2}
2388:
2387:
2327:
2326:
2318:
2228:, order 384
2190:
2052:
2051:
1979:Constituent
1968:
1967:
1962:
1948:
1891:
1860:
1749:enantiomorph
1201:
1194:
1179:
1177:
1163:
1152:
1145:
1137:
1133:
1111:Dodecahedron
1072:
1068:
1059:Dodecahedron
1007:
1003:
967:Dodecahedron
958:
954:
909:
905:
832:
823:
821:
799:(light) and
778:(light) and
757:(light) and
732:
728:
724:
720:
716:
712:
708:
704:
700:
696:
692:
688:
684:
680:
676:
672:
668:
664:
660:
656:
652:
648:
644:
640:
636:
632:
628:
624:
620:
616:
612:
608:
604:
600:
596:
592:
588:
584:
580:
574:
569:
567:
564:
554:The regular
553:
526:
515:
511:
502:
498:
491:
490:Icosahedron
486:
451:
447:
438:
434:
427:
420:
388:
379:
375:
369:Icosahedron
365:
326:
318:
311:
304:
302:Dodecahedron
269:
265:
256:
252:
240:
208:constituent
192:Common core
145:
130:
121:
102:
75:
56:
50:such as the
35:
29:
3290:Dual Models
3265:, Cambridge
2804:, 2 {3,3}.
2755:orbit space
2454:5 {5,3,5/2}
2451:5 {5/2,3,5}
2432:5 {3,5/2,5}
2429:5 {5,5/2,3}
2410:5 {5/2,5,3}
2407:5 {3,5,5/2}
2399:Compound 2
2396:Compound 1
2378:{5/2,5,5/2}
2367:{5/2,5,5/2}
2063:Compound 2
2060:Compound 1
2053:Dual pairs:
2014:order 1200
1969:Self-duals:
1867:icosahedron
1124:Icosahedron
1046:Icosahedron
971:icosahedron
840:convex hull
811:Archimedean
521:Five cubes
309:Icosahedron
284:{5,3}{3,5}
220:{4,3}{3,4}
204:restricting
188:Convex hull
78:stellations
67:convex hull
65:called its
3210:References
2991:2020-09-03
2819:Self-dual
2813:Self-dual
2751:stabilizer
2490:Compound 2
2483:Compound 1
2313:order 600
2224:tesseracts
2204:Compound 2
2197:Compound 1
2164:tesseracts
2147:tesseracts
2134:order 600
2130:tesseracts
2113:tesseracts
2096:tesseracts
2079:tesseracts
1873:) and the
1413:octahedral
922:octahedron
898:Octahedron
869:tetrahedra
731:} counted
715:} counted
667:} counted
627:} counted
583:separate {
549:stellation
545:octahedron
529:tetrahedra
399:Five cubes
245:Octahedron
184:Spherical
57:The outer
3263:Polyhedra
3254:123279687
3008:(1973) .
2949:Footnotes
2597:tesseract
2577:2 24-cell
2556:Symmetry
2550:Compound
2527:{5/2,3,3}
2521:{3,3,5/2}
2510:{5/2,3,3}
2504:{3,3,5/2}
2497:Symmetry
2402:Symmetry
2356:{5,5/2,5}
2345:{5,5/2,5}
2338:Symmetry
2335:Compound
2292:120-cells
2286:600-cells
2275:120-cells
2269:600-cells
2211:Symmetry
2066:Symmetry
1982:Symmetry
1976:Compound
1898:pentagram
1896:, as the
1184:prismatic
1122:(Convex:
1109:(Convex:
1057:(Convex:
1044:(Convex:
828:midsphere
776:Snub cube
112:talk page
71:facetting
40:polyhedra
3405:Category
3330:(1509),
3288:(1983),
3271:citation
2885:3 {3,6}
2882:3 {6,3}
2619:600-cell
2615:120-cell
2562:2 5-cell
2309:24-cells
2303:24-cells
2258:24-cells
2252:24-cells
2241:24-cells
2235:24-cells
2218:16-cells
2181:24-cells
2175:24-cells
2158:16-cells
2141:16-cells
2124:16-cells
2107:16-cells
2090:16-cells
2073:16-cells
2037:24-cells
1959:McMullen
851:Picture
201:Subgroup
181:Picture
105:disputed
59:vertices
52:hexagram
32:geometry
3246:0397554
3226:Bibcode
2792:} (4 ≤
2749:is the
2601:16-cell
2582:24-cell
2042:24-cell
2021:5-cells
2005:5-cells
1989:5-cells
1951:Coxeter
1942:{3,3,4}
1936:{4,3,3}
1747:68-75:
466:2{3,5}
402:2{5,3}
3390:
3362:
3347:
3321:: 5–97
3252:
3244:
3030:798003
3028:
3018:
2816:Duals
2567:5-cell
2026:5-cell
2010:5-cell
1994:5-cell
803:(dark)
782:(dark)
761:(dark)
719:times
593:before
537:Kepler
332:Chiral
206:to one
158:, and
44:centre
3250:S2CID
2890:{∞,∞}
2844:{∞,∞}
2838:{3,6}
2832:{6,3}
2826:{4,4}
2796:≤ ∞,
2737:, if
1751:pairs
857:Core
854:Hull
633:after
168:flags
150:, is
3388:ISBN
3360:ISBN
3345:ISBN
3277:link
3026:OCLC
3016:ISBN
2256:200
2250:200
2239:100
2233:100
2162:600
2156:600
2145:300
2139:300
2019:720
2003:120
1987:120
1881:and
1869:and
1157:and
1130:*532
1083:and
1065:*532
1018:and
1000:*532
969:and
951:*432
920:and
918:Cube
902:*432
893:Cube
867:Two
824:dual
813:and
541:cube
495:*532
431:*532
372:*532
334:twin
262:*332
249:*432
238:Cube
34:, a
3380:doi
3234:doi
2525:10
2519:10
2376:10
2354:10
2307:25
2301:25
2290:10
2284:10
2179:25
2173:25
2128:75
2122:75
2111:75
2105:75
2094:15
2088:15
1940:75
1934:75
1415:or
721:and
508:3*2
444:3*2
385:332
323:332
315:532
30:In
3407::
3386:,
3319:11
3317:,
3273:}}
3269:{{
3248:,
3242:MR
3240:,
3232:,
3222:79
3220:,
3098:^
3052:^
3024:.
2984:.
2957:^
2888:3
2842:2
2836:2
2830:2
2824:2
2766:gH
2717:,
2699:,
2681:,
2663:,
2617:,
2599:,
2508:5
2502:5
2477::
2365:5
2343:5
2273:5
2267:5
2222:2
2216:2
2077:3
2071:3
2035:5
1965:.
1957:.
1904:.
1150:.
877:,
842:.
822:A
572:.
154:,
80:.
54:.
3397:.
3382::
3368:.
3335:.
3323:.
3306:.
3294:.
3281:.
3279:)
3257:.
3236::
3228::
3204:.
3142:.
3127:.
3046:.
3032:.
2994:.
2798:p
2794:p
2790:p
2788:,
2786:p
2770:g
2762:H
2760:/
2758:G
2747:H
2739:G
1419:,
1358:,
1204:)
1138:h
1134:I
1126:)
1113:)
1093:)
1089:(
1073:h
1069:I
1061:)
1048:)
1028:)
1024:(
1008:h
1004:I
979:)
975:(
959:h
955:O
930:)
926:(
910:h
906:O
881:)
873:(
733:e
729:t
727:,
725:s
717:c
713:n
711:,
709:m
705:q
703:,
701:p
699:{
697:d
693:t
691:,
689:s
687:{
685:e
683:}
681:n
679:,
677:m
675:{
673:c
669:e
665:t
663:,
661:s
657:q
655:,
653:p
651:{
649:d
645:t
643:,
641:s
639:{
637:e
629:c
625:n
623:,
621:m
617:q
615:,
613:p
611:{
609:d
605:n
603:,
601:m
599:{
597:c
589:q
587:,
585:p
581:d
516:h
512:T
503:h
499:I
452:h
448:T
439:h
435:I
389:T
380:h
376:I
327:T
319:I
270:d
266:T
257:h
253:O
137:)
131:(
126:)
122:(
118:.
108:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
